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Triangle Proof

- by Kathy McDonald
- section 3.1 7

Prove When dividing each side of an equilateral

triangle

into n segments

then connecting the division points with all

possible segments parallel

to the original sides, n² small triangles are

created.

Proof by induction

Let S n?N f(n) n²

Show 1 ?S

1

f(n) n² f(1) 1 1²

Show 2 ?S

when dividing each side into 2 segments

and connecting division points as described,

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4 small triangles are created.

f(n) n² f(2) 4 2²

Show 3 ?S

when dividing each side into 3 segments

and connecting division points as described,

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9 small triangles are created.

f(n) n² f(3) 9 3²

Assume n ?S.

Assume when dividing each side into n segments

and connecting division points as described, n²

small triangles are created. Assume f(n) n².

Show n1 ?S.

Show when dividing each side into n1 segments

and connecting division points as described,

(n1)² small triangles are created. Show f(n1)

(n1)².

Consider a divided triangle

with n segments on each side.

When a segment equal in size to the n segments is

added to each side

and those endpoints are connected,

a space is created at the bottom of the original

triangle.

Also, a new, bigger equilateral triangle has

been created.

This new, bigger triangle has n1 segments on

each side.

n segments

1 segment

Now, the parallel dividing lines are extended

down

to the base of the new, bigger triangle.

More small triangles are created.

The n segments of the base of the original

triangle

correspond to n bases of the new, small triangles

created.

Also, the n1 segments of the base of the new,

bigger triangle

correspond to n1 bases of the new, small

triangles.

So, n(n1) bases

correspond to n(n1) new, small triangles

By assumption, the original triangle has n

segments on each side

And n² small triangles inside.

By adding 1 segment to each side of this triangle,

n (n1) small triangles are added.

The total small triangles of the new, bigger

triangle is

n² n (n1)

n²2n1 (n1)(n1)

(n1)²

This shows n1 ?S.

By induction, S ? N.

Dwight says, thats it.